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Understanding the Expression (9+x)(9-x)
The expression (9+x)(9-x) is a special case of a common algebraic pattern known as the difference of squares. This pattern is a crucial concept in simplifying algebraic expressions and solving equations.
The Difference of Squares Pattern
The difference of squares pattern states that:
(a + b)(a - b) = a² - b²
In other words, the product of the sum and difference of two terms is equal to the square of the first term minus the square of the second term.
Applying the Pattern to (9+x)(9-x)
In our case, a = 9 and b = x. Therefore, we can apply the difference of squares pattern to simplify the expression:
(9+x)(9-x) = 9² - x² = 81 - x²
Significance of the Result
The simplified expression 81 - x² highlights the power of the difference of squares pattern. It allows us to quickly expand and simplify the original expression without needing to perform the full multiplication. This simplification can be very useful in solving equations, factoring expressions, and simplifying complex mathematical problems.
Examples and Applications
Here are some examples of how the difference of squares pattern is used in various contexts:
- Factoring expressions: If you need to factor the expression 81 - x², you can directly apply the pattern and get (9+x)(9-x).
- Solving equations: If you have an equation like 81 - x² = 0, you can factor it using the difference of squares pattern to find the solutions x = 9 and x = -9.
- Simplifying expressions: The difference of squares pattern can simplify complex expressions involving squares and differences, allowing for easier analysis and manipulation.
The difference of squares pattern is a fundamental concept in algebra. By understanding this pattern and its applications, you can efficiently simplify expressions, solve equations, and gain a deeper understanding of algebraic manipulations.